Holonomy Guide, Meaning , Facts, Information and Description
In differential geometry, the holonomy of a given structure (for example a Riemannian metric, or more general G-structure) at a point P on a smooth manifold M is the group of all linear maps transforming the tangent space at P that can be induced by a parallel transport around a loop based at P. A generic d-dimensional Riemannian manifold has an O(d) holonomy, or SO(d), if it is orientable. A flat d-dimensional manifold (such as the torus) has as holonomy a discrete group.
The Berger list from 1955 was a list of holonomy groups possible in the case of an affine connection, which allowed for a finite number of further exceptional cases. Research from 1995 showed some modification was necessary.
Intermediate cases are the most interesting ones. Consider Calabi-Yau manifolds: SU(3) holonomy is the maximum holonomy of six-dimensional (three-complex-dimensional) Calabi-Yau manifolds. A generic six-dimensional manifold has an O(6) holonomy (or SO(6), if it is orientable). A flat six-dimensional manifold (such as the six-torus) has a discrete holonomy. SU(3) is something in between; a subgroup of SU(4) (which is locally isomorphic to SO(6)) that guarantees that one quarter of the original supersymmetry will be preserved if a manifold of SU(3) holonomy is used for compactification of string theory. A generic Kähler manifold would have a U(3) holonomy; and the component of the holonomy group which contains the identity is restricted to be in the SU(3) group if the manifold is Ricci flat.
Calabi-Yau d-folds have an SU(d) holonomy and they are a special case of Kähler manifolds with U(d) holonomy. There exist seven-dimensional real manifolds with holonomy G2 - the so-called G2 manifolds - as well as eight-dimensional real manifolds with holonomy , hyperkähler manifolds with holonomy — the real dimension must be a multiple of four in this case — and these hyperkähler manifolds are a special class of quaternion Kähler manifolds with holonomy locally isomorphic to .
Homogeneous spaces can also have holonomy locally isomorphic to , and the holonomies can be combined into a direct product for manifolds that have the form of a Cartesian product of two "smaller" manifolds. The description above lists all possibilities, though only in the simply connected case, as the Berger list shows.
The local holonomy group is the subgroup of the holonomy group that only contains the transformations induced by the parallel transport along contractible loops. Therefore, the notions of holonomy and local holonomy are equivalent for simply connected manifolds.